They are jump infantry - by definition jump infantry can deep strike.
Yes, they can deep strike.
It is impossible to shoot/run/assault - by definition, to run, you give up the chance to shoot. They can move/run/assault though because they have Fleet.
They do not have the relentless rule...but they also can't take any weapons that would be helped by the relentless rule, so I'm not sure that it matters. They can fire their splinter pods normally after moving 12" though, if that was your question.